Metric Geometry of Groups and Surfaces

群和曲面的度量几何

• 批准号: 0906086
• 负责人: Moon Duchin
• 金额: $ 10.71万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906086&HistoricalAwards=false
• 关键词: Metric; Geometry; Groups; Surfaces

AbstractAward: DMS-0906086Principal Investigator: Moon DuchinThis award is funded under the American Recovery and Reinvestment Act of 2009(Public Law 111-5).The projects in this proposal are clustered around themes inlow-dimensional topology, asymptotic geometry, and the dynamicsof group actions. Teichmuller space and the mapping class groupare objects for special attention, and tools developed for thatsetting are often promising for wider study in metric geometryand geometric group theory. The PI proposes to build on recentwork on flat metrics, their length spectra, and the associatedboundary theory, as well as work on synthetic curvatureconditions and their applications to random walks. Specificavenues of future investigation include asymptotics on the spaceof singular flat surfaces; filling inequalities and higherdivergence functions adapted from symmetric spaces to grouptheory; and the use of intersection patterns of metrichalf-spaces to study the dynamics of isometries.While quite abstract, ideas in this work can also provebeautifully applicable-- for instance, Thurston's theory of flowson surfaces has direct and strong applications to the study ofmixing in fluid dynamics. More immediately, though, thisresearch area addresses foundational questions towardunderstanding geometric structure of complicated spaces in thelarge, in the small, and changing over time.

奖项：dms -0906086首席研究员：Moon duchin2009美国复苏与再投资法案（公法111-5）资助。本提案中的项目围绕低维拓扑、渐近几何和群体行动的动态主题聚集。Teichmuller空间和映射类群是值得特别关注的对象，为此开发的工具通常有望在度量几何和几何群论中进行更广泛的研究。PI建议建立在最近关于平面度量，它们的长度谱和相关边界理论的工作，以及关于合成曲率条件及其在随机漫步中的应用的工作。未来研究的具体方向包括奇异平面空间的渐近性；从对称空间到群论的填充不等式和高散度函数并利用度量半空间的交点模式来研究等距的动力学。虽然相当抽象，但这项工作中的思想也可以被证明是非常适用的——例如，瑟斯顿的流动表面理论对流体动力学中的混合研究有直接而有力的应用。然而，更直接的是，这个研究领域解决了理解复杂空间的几何结构的基本问题，无论是大的，小的，还是随时间变化的。